Problem: Starting at home, Emily traveled uphill to the gift store for 12 minutes at just 10 mph. She then traveled back home along the same path downhill at a speed of 30 mph. What is her average speed for the entire trip from home to the gift store and back?
Explanation: The average speed is not just the average of 10 mph and 30 mph. She traveled for a longer time uphill (since she was going slower), so we can estimate that the average speed is closer to 10 mph than 30 mph. To calculate the average speed, we will make use of the following: $\text{average speed} = \dfrac{{\text{total distance}}}{{\text{total time}}}$ $\text{distance uphill} = \text{distance downhill}$ What was the total distance traveled? ${\begin{align*}\text{total distance} &= \text{distance uphill} + \text{distance downhill}\\ &= 2 \times \text{distance uphill}\end{align*}}$ $\begin{align*}\text{distance uphill} &= \text{speed uphill} \times \text{time uphill} \\\ &= 10\text{ mph} \times 12\text{ minutes}\times\dfrac{1 \text{ hour}}{60 \text{ minutes}}\\ &= 2\text{ miles}\end{align*}$ Substituting to find the total distance: ${\text{total distance} = 4\text{ miles}}$ What was the total time spent traveling? ${\text{total time} = \text{time uphill} + \text{time downhill}}$ $\begin{align*}\text{time downhill} &= \dfrac{\text{distance downhill}}{\text{speed downhill}}\\ &= \dfrac{2\text{ miles}}{30\text{ mph}}\times\dfrac{60 \text{ minutes}}{1 \text{ hour}}\\ &= 4\text{ minutes}\end{align*}$ ${\begin{align*}\text{total time} &= 12\text{ minutes} + 4\text{ minutes}\\ &= 16\text{ minutes}\end{align*}}$ Now that we know both the total distance and total time, we can find the average speed. $\begin{align*}\text{average speed} &= \dfrac{{\text{total distance}}}{{\text{total time}}}\\ &= \dfrac{{4\text{ miles}}}{{16\text{ minutes}}}\times\dfrac{60 \text{ minutes}}{1 \text{ hour}}\\ &= 15\text{ mph}\end{align*}$ The average speed is 15 mph, and which is closer to 10 mph than 30 mph as we expected.